402 Practical Mathematics

238. The equations (i), (2), (3) are all of the form

........ <s>

where a and b are constants.

We thus find that there are three forms of the solution.
If the quadratic

A.2 4. ax + b = o

has two unequal roots \lt A.2J the solution is in the form
y = Aj^i* + A2<?X2* as in (5)
If the quadratic has imaginary roots the solution takes the form

y - A*** sin (#*• -i- B) as in (6)
where the roots of the quadratic are

* m,± *. =

and therefore the solution is

..... (9)

If the quadratic has equal roots \, the solution is y = (A 4-
The student is advised not to use the general formulae, but to work out
each case separately as has been done for equations (i), (2), (3) above.

237. Simple Harmonic Motion. — An important special case arises
when a — o in equation (8).
We get

The solution (9) becomes

y = A sin (jUx + B)

The case of simple harmonic motion is represented by a differential
equation of this form.

Let a point move along a straight line so that its acceleration towards a
fixed point O is proportional to its distance from that point.

Let x be the distance of the moving point from O at any time /.

Then the above condition gives

— = — q*x where q is a constant

d*x
Or ~dP + ?* = °

The solution of this differential equation is

y = A sin (gt + g)

where g is a constant.

This gives the distance x described in time / in simple harmonic motion,-
This is the same as the equation obtained to represent simple harmonic